5 Answers
5

No, you only roll 1 additional die. The player's handbook is consistent in its use of the words 'die' and 'dice', where die is singular and 'dice' is singular or plural. A good example of this is on page 196 under Damage Rolls:

You roll the damage die or dice, add any modifiers, and apply the damage to your target.

There is also this quote that wotc_rodney tweeted (thanks GMNoob) which states that it is inteded to only be 1 die:

Also anyone that gets to roll one extra die on crits will favor the greataxe (like barbarians).

This reading produces the strange result that a greatsword deals only slightly more damage than a two-handed longsword: 5d6 (17.5) versus 3d10 (16.5). That doesn't seem like an intended result.
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Bradd SzonyeAug 24 '14 at 11:40

@BraddSzonye Strange is personal judgement and doesn't equal "must be unintended by the desingers" though. Regardless of whether the conclusion is correct or not, the reasoning is invalid as it's a logical leap, with the leap bridged by personal preference.
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SevenSidedDieAug 24 '14 at 21:17

5

The quote suggests that this is to make axes more attractive than swords, but it also makes smaller swords relatively much better than a big sword, which is just kind of bizarre. That makes it sound like the quote is a spur of the moment rationalization rather than a considered design decision, so I don't think it's good evidence in favor of the ruling.
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Bradd SzonyeAug 24 '14 at 21:23

2

@briddums The key word is relatively. In all other cases, the barbarian ability makes bigger = a lot better, but in the case of swords, biggest = hardly better at all. That’s counterintuitive and thus I doubt it’s intentional.
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Bradd SzonyeSep 5 '14 at 0:09

1

Depends on desire... you get consistently higher non-crit damage with a 2-handed sword when you reroll 1's and 2's because of Two-Handed Fighting Style. This makes the choice between GreatAxe and GreatSword very interesting. Do you want crits to be more powerful, or do you want average rolls to be more powerful? (With BattleMaster Fighter, I'd go GSword.)
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AvioseNov 20 '14 at 22:38

I don't see anywhere that it specifically defines "damage die" for a weapon, but the fact that you roll one extra die, then two and three, indicates to me that you should only roll one extra die (of same the number of sides as the damage die or dice of the weapon).

For a similar construction, see PHB 196 which describes rolling "the damage die or dice" of a weapon, spell, or harmful ability.

This is in contrast to 4e, which specifically says that the "W" of a weapon is equal to its total damage, so that "2W" would mean twice the number of dice.

Additional Justification

5e is a game where the rules try to homogenize a lot of roles. Most classes either use strong armor or have easy access to non-armor AC of 13 + a modifier. Everyone has the same proficiency bonus at the same level. Et cetera. The designers have called the general concept here "bounded accuracy," but I think it applies to damage as well.

If you roll 2d6 each time, then the multiplicative effect means that by the end you'd be rolling an extra 6d6 in comparison to barbarians using most weapons, who would be rolling an extra 3dX. This doesn't seem consistent with the even capabilities shown elsewhere in the system.

From a realism standpoint, my interpretation of the rule would mean that while a Greatsword does a ton of damage all the time, you can't squeeze much more extra damage out of it. There's a limit to how much harm you can cause, even as a lucky and skilled barbarian.

Final Note

Of course, this doesn't matter too much. It only happens on crit, and the expanded critical ranges come with Fighter, so it's only likely to be notably broken once you reach Fighter 3/Barb 9. 12th level is a good time to deal arbitrary amounts of damage.

This does seem like a logical parsing of the rules, but I have a hard time believing that it is intended. (I suspect it's just a side-effect of attempting to write the rules in natural language instead of as a technical code.)
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mattdmAug 23 '14 at 10:50

Other 2-handed barbarians would be rolling an extra 3d12...That's not a big difference from 6d6 (same max, different minimum, averages are 19.5 vs 21). So I'm not sure that's a particularly good argument. In contrast, if you don't get the second d6, you're rolling 3d6 vs 3d12, and that is a big difference (same min, much lower max, avg of 10.5 vs 19.5)
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wax eagleAug 23 '14 at 12:10

The "point" of weapons with multiple damage dice is increased consistency, not reduced total damage potential, which rolling only one of the dice would obviously be.

I would consider this to be a case of "specific beats general" — the general case of the casual language in the "one additional weapon damage die" is replaced by the corner case where a few weapons actually have more then one die for weapon damage.

I think the intent of the wording is that you only add this once, and not any other additional dice from any other source (for example, Sneak Attack damage, which is normally rolled a second time in 5e critical hits — you don't roll that yet again with this feature).

It would be completely reasonable for there to be some special, intentional rules reason to pick axes over swords, but I don't think it is meant to be found in basically-obscure loopholes. The only thing like that currently is in the barbarian starting equipment list, which suggests a greataxe and handaxes (but which also allows other martial weapons). It would have been nice to see a feat or barbarian class feature instead, but it does not appear to be there. (Maybe in future material, if they ever get around to producing 5E class "splatbooks".)

The one side effect of the other reading that would be useful, is that it would make the great axe a useful weapon. Right now there is literally no reason to take it over a greatsword. I favor this reading generally because it's the one that makes sense to me (why would you penalize someone for wielding a weapon with 2 "dice" as its "die"
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wax eagleAug 23 '14 at 11:58

@waxeagle: The same logic applies to a battleaxe vs a longsword; I suspect that the cheaper price of the base item may carry over to magical variants, as that's the only reason to choose a heavier axe over a more-expensive sword.
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DougMAug 23 '14 at 13:35

@waxeagle In a recent interview with mike mearls, (gamers with jobs) he said straight out that he is aware some people will think the greatsword is always better than the greataxe, but they preferd simplicity and flavor over equality when the difference is that small.
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GMNoobAug 23 '14 at 17:55

This makes weapons that do 1d12 more effective at criticals than the ones that do 2d6, which average a little more damage instead. I think this is done to differentiate the weapons. A criticals-focused fighter has more advantage with 1d12 weapons.

If instead the rules said that you could roll an extra 2d6, that would make the Great Axe almost useless, mathematically. In the 3.5 edition, for example, the Axe does 1d12 damage / ×3 criticals, while the Two-Handed Sword does 2d6/×2; in D&D 5e it is more or less the same: Axes are meant to be more "brutal" and a very good choice for Barbarians.